A073002 -id:A073002 - cp's OEIS frontend

A242051 Decimal expansion of B, a constant appearing in the asymptotic number of integers the prime factorization of which has decreasing exponents.

1, 8, 8, 7, 0, 2, 9, 9, 6, 5, 4, 3, 0, 8, 2, 5, 2, 7, 8, 2, 4, 8, 1, 3, 8, 1, 9, 6, 7, 9, 9, 5, 6, 9, 9, 1, 1, 5, 3, 7, 8, 6, 6, 2, 3, 8, 0, 8, 8, 4, 9, 9, 7, 8, 0, 3, 4, 8, 8, 3, 0, 4, 4, 7, 3, 8, 7, 0, 8, 9, 0, 9, 0, 5, 6, 0, 9, 1, 4, 2, 0, 5, 3, 2, 4, 6, 7, 2, 3, 9, 0, 5, 4, 9, 5, 6, 9, 0, 0, 2, 8, 9, 4, 8, 6

Offset: 1

Jean-François Alcover, Aug 13 2014

G. C. Greubel, Table of n, a(n) for n = 1..10000
Steven R. Finch, Errata and Addenda to Mathematical Constants. p. 9.
L. B. Richmond, Asymptotic results for partitions (I) and the distribution of certain integers, Journal of Number Theory 8, 372-389 (1976) p. 388.

Cf. A001620, A073002.

RealDigits[Zeta'[2] - (Pi^2/6)*EulerGamma, 10, 105] // First

default(realprecision, 100); zeta'(2) - zeta(2)*Euler \\ G. C. Greubel, Sep 06 2018

B = -integral_{y>0} log(1-e^(-y))*log(y) dy = zeta'(2) - (Pi^2/6)*gamma.

A271854 Decimal expansion of -zeta'(-1/2), negated derivative of the Riemann zeta function at -1/2.

Original entry on oeis.org

3, 6, 0, 8, 5, 4, 3, 3, 9, 5, 9, 9, 9, 4, 7, 6, 0, 7, 3, 4, 7, 4, 2, 0, 8, 0, 6, 3, 6, 3, 9, 5, 1, 0, 6, 5, 8, 8, 4, 8, 5, 2, 7, 8, 7, 9, 1, 8, 6, 3, 2, 2, 1, 0, 8, 1, 4, 3, 7, 6, 2, 8, 1, 2, 7, 5, 8, 0, 8, 1, 0, 6, 1, 2, 6, 6, 5, 6, 5, 1, 0, 3, 0, 9, 5, 7, 3, 3, 0, 8, 5, 0, 8, 3, 0, 9, 1, 6, 0, 2, 8, 5, 0, 8, 1

Offset: 0

Stanislav Sykora, Apr 23 2016

			zeta'(-1/2) = -0.36085433959994760734742080636395106588485278791863221...

Stanislav Sykora, Table of n, a(n) for n = 0..1000

Values of |zeta'(x)| for various x: A073002 (+2), A075700 (0), A084448 (-1), A114875 (+1/2), A240966 (-2), A244115(+3), A259068 (-3), A259069 (-4), A259070 (-5), A259071 (-6), A259072 (-7), A259073 (-8), A261506 (+4), A266260 (-9), A266261 (-10), A266262 (zeta'(-11)), A266263 (zeta'(-12)), A260660 (zeta'(-13)), A266264 (zeta'(-14)), A266270 (zeta'(-15)), A266271 (zeta'(-16)), A266272 (zeta'(-17)), A266273 (zeta'(-18)), A266274 (zeta'(-19)), A266275 (zeta'(-20)), A271521 (i).

RealDigits[N[-Zeta'[-1/2], 106]] [[1]] (* Robert Price, Apr 28 2016 *)

PARI
```
-zeta'(-1/2)
```

A273240 Decimal expansion of Integral_{0..inf} x log(x)/(exp(x)-1) dx (negated).

Original entry on oeis.org

2, 4, 2, 0, 9, 5, 8, 9, 8, 5, 8, 2, 5, 9, 8, 8, 4, 1, 7, 7, 5, 7, 2, 3, 0, 3, 0, 1, 5, 3, 5, 4, 4, 7, 2, 2, 3, 1, 8, 9, 1, 6, 3, 3, 6, 8, 8, 1, 7, 0, 1, 3, 4, 2, 6, 1, 3, 2, 7, 2, 2, 1, 8, 0, 1, 7, 0, 8, 1, 6, 2, 0, 1, 5, 7, 7, 1, 3, 3, 3, 1, 4, 9, 1, 0, 4, 3, 4, 8, 9, 9, 2, 9, 8, 1, 0, 2, 9, 7, 5, 9

Offset: 0

Jean-François Alcover, May 18 2016

			-0.242095898582598841775723030153544722318916336881701342613272218...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Donal F. Connon, Some series and integrals involving the Riemann zeta function, binomial coefficients and the harmonic numbers. Volume II(b), arXiv:0710.4024 [math.HO] 2007. page 130.

Cf. A001620, A073002, A074962.

RealDigits[(1/6) Pi^2 (1 + Log[2Pi] - 12 Log[Glaisher]), 10, 101][[1]]

default(realprecision, 100); (1/6)*(1-Euler)*Pi^2 + zeta'(2) \\ G. C. Greubel, Sep 07 2018

Equals (1/6)*(1-EulerGamma)*Pi^2+zeta'(2).

Also equals (1/6)*Pi^2*(1+log(2*Pi)-12*log(G)), where G is the Glaisher-Kinkelin constant.

A335006 Decimal expansion of gamma - zeta'(2)/zeta(2), where gamma is the Euler-Mascheroni constant.

Original entry on oeis.org

1, 1, 4, 7, 1, 7, 6, 6, 5, 7, 9, 9, 6, 0, 6, 5, 6, 6, 7, 0, 0, 6, 3, 7, 6, 4, 5, 0, 1, 0, 2, 1, 3, 2, 4, 3, 3, 4, 4, 5, 6, 4, 1, 6, 1, 6, 7, 4, 6, 8, 5, 4, 5, 7, 8, 3, 6, 3, 9, 6, 4, 5, 9, 5, 3, 2, 2, 7, 8, 5, 0, 0, 4, 1, 4, 4, 4, 0, 5, 2, 7, 7, 7, 2, 4, 8, 1

Offset: 1

Amiram Eldar, May 19 2020

			1.1471766579960656670063764501021324334456416167468...

Eckford Cohen, Arithmetical Functions of a Greatest Common Divisor. I, Proceedings of the American Mathematical Society, Vol. 11, No. 2 (1960), pp. 164-171.

Cf. A000010 (phi), A001620 (gamma), A013661 (zeta(2)), A073002 (-zeta'(2)).

RealDigits[EulerGamma - Zeta'[2]/Zeta[2], 10, 100][[1]]

Euler - zeta'(2)/zeta(2) \\ Michel Marcus, May 19 2020

Equals lim_{k->oo} zeta(2) * Sum_{i=1..k} phi(i)/i^2 - log(k), where phi is the Euler totient function (A000010).

Equals A001620 + A073002/A013661.

A343481 a(n) is the sum of all digits of n in every prime base 2 <= p <= n.

Original entry on oeis.org

1, 3, 3, 6, 6, 10, 11, 11, 10, 15, 16, 22, 21, 21, 23, 30, 32, 40, 42, 42, 39, 48, 52, 53, 49, 52, 53, 63, 66, 77, 83, 82, 76, 77, 82, 94, 87, 85, 90, 103, 107, 121, 123, 129, 120, 135, 144, 147, 153, 150, 151, 167, 176, 178, 185, 181, 168, 185, 194, 212, 199

Offset: 2

Amiram Eldar, Apr 16 2021

			a(5) = 6 since in the prime bases 2, 3 and 5 the representations of 5 are 101_2, 12_3 and 10_5, respectively, and (1 + 0 + 1) + (1 + 2) + (1 + 0) = 6.

Amiram Eldar, Table of n, a(n) for n = 2..10001
Robin Fissum, Digit sums and the number of prime factors of the factorial n!=1.2...n, Bachelor's project in BMAT, Norwegian University of Science and Technology, 2020; ResearchGate link.

Cf. A014837, A043306, A072691, A073002, A222171.

s[n_, b_] := Plus @@ IntegerDigits[n, b]; ps[n_] := Select[Range[n], PrimeQ]; a[n_] := Sum[s[n, b], {b, ps[n]}]; Array[a, 100, 2]

a(n) = sum(b=2, n, if (isprime(b), sumdigits(n, b))); \\ Michel Marcus, Apr 17 2021

a(n) ~ (1-Pi^2/12)*n^2/log(n) + c*n^2/log(n)^2 + o(n^2/log(n)^2), where c = 1 - Pi^2/24 + zeta'(2)/2 = 1 - A222171 - (1/2)*A073002 = 0.1199923561... (Fissum, 2020).

A345726 a(n) = Product_{k=1..n} k^(floor(n/k)^2).

Original entry on oeis.org

1, 2, 6, 192, 960, 4976640, 34836480, 2283043553280, 4993016251023360, 3195530400654950400000, 35150834407204454400000, 417877827219530751882239882035200000, 5432411753853899774469118466457600000, 213700126654516647665669790727613605478400000

Offset: 1

Vaclav Kotesovec, Jun 25 2021

nonn

G. C. Greubel, Table of n, a(n) for n = 1..55

Cf. A073002, A092143, A129364.

[(&*[j^(Floor(n/j))^2: j in [1..n]]): n in [1..30]]; // G. C. Greubel, Feb 05 2024

Table[Product[k^(Floor[n/k]^2), {k, 1, n}], {n, 1, 15}]

a(n) = prod(k=1, n, k^((n\k)^2)); \\ Michel Marcus, Jun 26 2021

[product(j^((n//j)^2) for j in range(1,n+1)) for n in range(1,31)] # G. C. Greubel, Feb 05 2024

log(a(n)) ~ c * n^2, where c = -zeta'(2) = A073002.

A349694 Dirichlet convolution of the squarefree kernel function (A007947) with itself.

Original entry on oeis.org

1, 4, 6, 8, 10, 24, 14, 12, 15, 40, 22, 48, 26, 56, 60, 16, 34, 60, 38, 80, 84, 88, 46, 72, 35, 104, 24, 112, 58, 240, 62, 20, 132, 136, 140, 120, 74, 152, 156, 120, 82, 336, 86, 176, 150, 184, 94, 96, 63, 140, 204, 208, 106, 96, 220, 168, 228, 232, 118, 480

Offset: 1

Ilya Gutkovskiy, Nov 25 2021

Cf. A007947, A097945, A176345, A191750.

Table[Sum[Last[Select[Divisors[d], SquareFreeQ]] Last[Select[Divisors[n/d], SquareFreeQ]], {d, Divisors[n]}], {n, 1, 60}]
f[p_, e_] := (e - 1)*p^2 + 2*p; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 60] (* Amiram Eldar, Nov 25 2021 *)

A007947(n) = factorback(factorint(n)[, 1]); \\ From A007947
A349694(n) = sumdiv(n,d,A007947(n/d)*A007947(d)); \\ Antti Karttunen, Nov 25 2021

Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 + p^(1-s) - p^(-s))^2.
a(n) = Sum_{d|n} A007947(d) * A007947(n/d).
a(n) = Sum_{d|n} abs(A097945(d)) * A191750(n/d).
Multiplicative with a(p^e) = (e-1)*p^2 + 2*p. - Amiram Eldar, Nov 25 2021
From Vaclav Kotesovec, Nov 26 2021: (Start)
Dirichlet g.f.: zeta(s-1)^2 * zeta(s)^2 * Product_{primes p} (1 + p^(1-2*s) - p^(2-2*s) - p^(-s))^2.
Let f(s) = Product_{primes p} (1 + p^(1-2*s) - p^(2-2*s) - p^(-s)), then
Sum_{k=1..n} a(k) ~ Pi^2 * f(2)^2 * n^2 / 144 * (Pi^2 * (2*log(n) + 4*gamma - 1 + 4*f'(2)/f(2)) + 24*zeta'(2)), where f(2) = Product_{primes p} (1 - 2/p^2 + 1/p^3) = A065464 = 0.428249505677094440218765707581823546121298513355936144..., f'(2) = f(2) * Sum_{primes p} log(p) * (3*p - 2) / (p^3 - 2*p + 1) = 0.6293283828324697510445630056425352981207558777167836747744750359407300858..., zeta'(2) = -A073002 and gamma is the Euler-Mascheroni constant A001620. (End)

A365207 The number of divisors d of n such that gcd(d, n/d) is a power of 2 (A000079).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 2, 4, 2, 6, 2, 4, 4, 5, 2, 4, 2, 6, 4, 4, 2, 8, 2, 4, 2, 6, 2, 8, 2, 6, 4, 4, 4, 6, 2, 4, 4, 8, 2, 8, 2, 6, 4, 4, 2, 10, 2, 4, 4, 6, 2, 4, 4, 8, 4, 4, 2, 12, 2, 4, 4, 7, 4, 8, 2, 6, 4, 8, 2, 8, 2, 4, 4, 6, 4, 8, 2, 10, 2, 4, 2, 12, 4, 4

Offset: 1

Amiram Eldar, Aug 26 2023

The sum of these divisors is A107749(n).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A000079, A000265, A007814, A034444, A038838, A042968, A107749, A122132.

Cf. A001620, A013661, A073002.

f[p_, e_] := If[p == 2, e + 1, 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1] == 2, f[i,2]+1, 2));}

Multiplicative with a(2^e) = e+1 and a(p^e) = 2 for an odd prime p.
a(n) <= A000005(n), with equality if and only if n is in A122132 (or equivalently, n is not in A038838).
a(n) >= A034444(n), with equality if and only if n is not divisible by 4 (A042968).
a(n) = A000005(A006519(n)) * A034444(A000265(n)).
a(n) = A034444(n) * (A007814(n)+1)/2^(1 - (n mod 2)).
Dirichlet g.f.: (4^s/(4^s-1)) * zeta(s)^2/zeta(2*s).
Sum_{k==1..n} a(k) ~ (8/Pi^2)*n*(log(n) + 2*gamma - 2*log(2)/3 - 2*zeta'(2)/zeta(2) - 1), where gamma is Euler's constant (A001620).

A375368 Decimal expansion of zeta'(2)/(2Pi^2) + log(2Pi)/6 - gamma/12.

Original entry on oeis.org

2, 1, 0, 7, 1, 4, 7, 8, 9, 5, 6, 8, 5, 5, 2, 1, 0, 8, 3, 4, 2, 9, 1, 1, 8, 7, 4, 6, 2, 6, 6, 9, 4, 8, 4, 3, 8, 3, 3, 3, 2, 9, 0, 2, 3, 1, 5, 0, 3, 5, 6, 5, 8, 9, 4, 0, 8, 7, 2, 0, 1, 3, 0, 5, 5, 0, 6, 8, 9, 8, 1, 4, 9, 6, 3, 7, 1, 9, 6, 9, 2, 7, 5, 4, 5, 1, 3, 2, 1

Offset: 0

R. J. Mathar, Aug 13 2024

zeta'(2) = -0.9375.. is the first derivative of the zeta function (see A073002). Gamma is A001620.

			0.21071478956855210834291187462669484383332902315035...

Olivier Espinosa and Victor H. Moll, On some integrals involving the Hurwitz zeta function: Part 1, Raman. J. 6 (2002) 159-188, Example 6.4.

Cf. A001620, A073002, A367842, A375369.

Zeta(1,2)/2/Pi^2+log(2*Pi)/6-gamma/12 ; evalf(%) ;

RealDigits[Zeta'[2] / (2*Pi^2) + Log[2*Pi] / 6 - EulerGamma / 12, 10, 120][[1]] (* Amiram Eldar, Aug 19 2024 *)

Equals Integral_{x=0..1} x* log(Gamma(x)) dx.

Equals log(A367842). - Hugo Pfoertner, Aug 19 2024

A375369 Decimal expansion of zeta'(2)/(2Pi^2) + zeta(3)/(4Pi^2) + log(2*Pi)/12 -gamma/12.

Original entry on oeis.org

0, 8, 8, 0, 0, 6, 8, 2, 4, 4, 2, 6, 1, 6, 6, 5, 8, 8, 8, 2, 6, 4, 4, 1, 7, 8, 2, 3, 6, 3, 5, 8, 0, 0, 1, 3, 8, 3, 6, 7, 6, 3, 2, 6, 1, 0, 8, 9, 0, 3, 3, 2, 9, 0, 1, 9, 2, 1, 6, 6, 7, 6, 3, 6, 6, 2, 6, 0, 0, 0, 1, 6, 9, 2, 0, 7, 7, 9, 8, 5, 8, 4, 8, 3, 1, 8, 3

Offset: 0

R. J. Mathar, Aug 13 2024

zeta'(2)= -0.9375.. is the first derivative of the zeta function, see A073002. gamma is A001620.

			0.08800682442616658882644178236358001383676326108903...

Olivier Espinosa and Victor H. Moll, On some integrals involving the Hurwitz zeta function: Part 1, Raman. J. 6 (2002) 159-188, Example 6.4.

Cf. A001620, A073002, A375368.

Zeta(1,2)/2/Pi^2+Zeta(3)/4/Pi^2+log(2*Pi)/12-gamma/12 ; evalf(%) ;

RealDigits[Zeta'[2] / (2*Pi^2) + Zeta[3] / (4*Pi^2) + Log[2*Pi] / 12 - EulerGamma / 12, 10, 120, -1][[1]] (* Amiram Eldar, Aug 19 2024 *)

Equals Integral_{x=0..1} x^2* log(Gamma(x)) dx.

cp's OEIS Frontend

A242051 Decimal expansion of B, a constant appearing in the asymptotic number of integers the prime factorization of which has decreasing exponents.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A271854 Decimal expansion of -zeta'(-1/2), negated derivative of the Riemann zeta function at -1/2.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A273240 Decimal expansion of Integral_{0..inf} x log(x)/(exp(x)-1) dx (negated).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A335006 Decimal expansion of gamma - zeta'(2)/zeta(2), where gamma is the Euler-Mascheroni constant.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A343481 a(n) is the sum of all digits of n in every prime base 2 <= p <= n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A345726 a(n) = Product_{k=1..n} k^(floor(n/k)^2).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

SageMath

Formula

A349694 Dirichlet convolution of the squarefree kernel function (A007947) with itself.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A365207 The number of divisors d of n such that gcd(d, n/d) is a power of 2 (A000079).

Views

Author

Keywords

Comments

A375368 Decimal expansion of zeta'(2)/(2Pi^2) + log(2Pi)/6 - gamma/12.

A375369 Decimal expansion of zeta'(2)/(2Pi^2) + zeta(3)/(4Pi^2) + log(2*Pi)/12 -gamma/12.